N4 Mathematics

Mathematics Downloaded from hlayiso.com N4

Module 1: Equations, manipulations and word problems Factorise 𝒂𝒂𝟑𝟑 ± 𝒃𝒃𝟑𝟑 To solve, let 𝑎𝑎 and 𝑏𝑏 be real numbers, variables, or algebraic expressions. 1. Write as sum or difference of two cubes. 2. Write in the factored form. 3. Simplify the factored form Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Equations, manipulations and word problems (continued) THE BASIC OPERATIONS &THE LAWS OF INDICES AND LOGARITHMS Solving exponential formulas: Solving logarithmic formulas: 1. Isolate the exponential expression. 1. Isolate the logarithmic term. 2. Apply the logarithm to each side of 2. Transform the logarithmic the formula, and then remove the formula to an exponential variable in the exponent by using formula. Logarithmic Laws. 3. Solve for the variable. 3. Solve for theDownloaded variable. from hlayiso.com www.futuremanagers.com

Module 1: Equations, manipulations and word problems (continued) SOLVE THREE SIMULTANEOUS LINEAR EQUATIONS FOR THREE UNKNOWN VARIABLES Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Equations, manipulations and word problems (continued) SOLVE REAL-LIFE WORD PROBLEMS IN EQUATIONS When an answer is needed to a mathematical problem that is described in words as well as numbers, it is called a word problem. You find the solution by setting up an equation that represents the problem, and then solve it. Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Determinants WRITE SIMULTANEOUS EQUATIONS IN DETERMINANT NOTATION AND EVALUATE BY APPLYING SARRUS’ RULE First write the equations in the standard form. The coefficients of the variables will be captured to form the determinants. Then, a second and third order determinant will be evaluated using Sarrus’ rule: Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Determinants (continued) IDENTIFY AND CALCULATE THE MINOR OF A THIRD ORDER DETERMINATION The Minor Expansion of a third order determinant requires the expansion of a row or column. In terms of the minor expansion, the third order determinant is defined as the sum of the product of the sign from the sign array for the specific entry, the entry and the minor of the entry. Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Determinants (continued) DEFINE AND DETERMINE THE COFACTOR OF THE MINOR The cofactor of a minor is given as: 𝐶𝐶𝑖𝑖𝑖𝑖 = (−1)𝑖𝑖+ . 𝑀𝑀𝑖𝑖𝑖𝑖 𝑗𝑗 where 𝑖𝑖 and 𝑗𝑗 are the respective row and column of the element, and 𝑀𝑀𝑖𝑖𝑖𝑖 the minor of the element. Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Determinants (continued) CALCULATE DETERMINANTS OF THE SECOND AND THIRD ORDER AND APPLY CRAMER’S RULE Cramer’s rule is an alternative method to solving a system of linear equations in three variables compared to the methods of substitution and elimination. Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Complex numbers WRITE COMPLEX NUMBERS IN THE RECTANGULAR (STANDARD) FORM A complex number can be represented in the rectangular form, that is 𝑧𝑧 = 𝑎𝑎 + 𝑏𝑏 𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are real numbers, and i = −1 is the imaginary unit with the property 𝑖𝑖 2 = −1. Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Complex numbers (continued) WRITE COMPLEX NUMBERS IN THE POLAR (TRIGONOMETRIC) FORM. A complex number can be represented in the polar form, that is 𝑧𝑧 = 𝑟𝑟(cos 𝜃𝜃 + 𝑖𝑖 sin 𝜃𝜃) where 𝑟𝑟 is the modulus, the distance from the origin to a point (𝑎𝑎; 𝑏𝑏) and 𝜃𝜃 is the argument, the positive angle measured from the positive real axis anticlockwise. Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Complex numbers (continued) SOLVE A QUADRATIC EQUATION WITH COMPLEX ROOTS Steps for solving a quadratic equation with complex roots: 1. Write the quadratic equation in the standard form, 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏 + 𝑐𝑐 = 0. 𝑏𝑏 2. Record the numeric values of a (coefficient of 𝑥𝑥 2 ), 𝑏𝑏 (coefficient of 𝑥𝑥) and 𝑐𝑐 (constant). 3. Substitute the numeric values of 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 into the quadratic formula, 𝑥𝑥 = −𝑏𝑏± 𝑏𝑏2 −4𝑎𝑎𝑎𝑎 and write the solutions in standard (rectangular) form. 2𝑎𝑎 Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Complex numbers (continued) SOLVE COMPLEX EQUATIONS To solve: 1. Write both sides of the complex equation in the standard (rectangular) form, 𝑎𝑎 + 𝑏𝑏 = 𝑐𝑐 + 𝑑𝑑 . 𝑏𝑏 𝑑𝑑 2. Equate the real parts, 𝑎𝑎 = 𝑐𝑐 and imaginary parts, 𝑏𝑏 = 𝑑𝑑. 3. Compute the unknown variables. 4. Verify the solution. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry APPLY THE CONCEPT OF NEGATIVE AND POSITIVE ANGLES TO ALL THE CALCULATIONS RELEVANT TO THE SYLLABUS Measuring angles from the positive 𝑥𝑥 −axis in a counter clockwise rotation yields a positive angle and the clockwise rotation yields a negative angle. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) APPLY THE IDENTITIES FOR 𝒔𝒔𝒔𝒔𝒔𝒔(𝒂𝒂 ± 𝒃𝒃), 𝒄𝒄𝒄𝒄 (𝒂𝒂 ± 𝒃𝒃), AND 𝒕𝒕 𝒄𝒄 𝒕𝒕𝒕𝒕 (𝒂𝒂 ± 𝒃𝒃). The sum and difference identities are also referred to as the compound angle or composite angle identities. Strategy for using the sum and difference identities: 1. Rewrite the angle as a compound angle, that is, 𝑎𝑎 ± 𝑏𝑏. 2. Expand the appropriate compound angle identity, that is, sine, cosine and tangent. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) 𝟏𝟏 DERIVE THE IDENTITIES FOR 𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔, 𝐜𝐜𝐜𝐜𝐜𝐜 𝟐𝟐𝟐𝟐 , 𝒕𝒕 𝒕𝒕 𝒏𝒏𝒏𝒏𝒏𝒏 , 𝒔𝒔𝒔𝒔 𝒏𝒏 𝟐𝟐 𝒙𝒙 = (𝟏𝟏 − 𝟐𝟐 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 ), 𝐜𝐜𝐜𝐜 𝐬𝐬 𝟐𝟐 𝒙𝒙 = (𝟏𝟏 + 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 ), 𝒔𝒔𝒔𝒔𝒔𝒔 𝒙𝒙 , 𝐜𝐜𝐜𝐜𝐜𝐜 𝒙𝒙 AND 𝒕𝒕 𝒕𝒕 𝒏𝒏 𝒙𝒙 AND APPLY 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 The double angle identities are derived from the compound angle identities. This gives: Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) DERIVE THE CO-RATIOS 𝒔𝒔𝒔𝒔𝒔𝒔(𝟗𝟗𝟗𝟗𝟗 ± ), 𝐜𝐜𝐜𝐜𝐜𝐜(𝟗𝟗𝟗𝟗𝟗 ± ), AND 𝐭𝐭𝐭𝐭𝐭𝐭(𝟗𝟗𝟗𝟗𝟗 ± ) 𝒙𝒙 𝒙𝒙 𝒙𝒙 AND APPLY The compound angle identities can also be used to derive co-function identities. This gives: Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) APPLY THE IDENTITIES TO SOLVE TRIGONOMETRIC EQUATIONS: 1. Rewrite the equation in terms of one trigonometric function if possible by using trigonometric identities. 2. Determine whether the equation is linear or quadratic. 3. Apply factorising techniques. 4. Solve the trigonometric equation and verify the solutions with the desired interval. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) APPLY THE IDENTITIES TO SIMPLIFY TRIGONOMETRIC EXPRESSIONS 1. Simplify the individual functions by using the trigonometric knowledge outlined. 2. Rewrite the simplified trigonometric functions in terms of sine and cosine. 3. Simplify the sine and cosine trigonometric functions. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) APPLY THE IDENTITIES TO PROVE TRIGONEMTRIC IDENTITIES 1. Identify the more complicated side. 2. Simplify the trigonometric functions. 3. Rewrite the simplified trigonometric functions in Step 2 in terms of sine and cosine. 4. Simplify the sine and cosine trigonometric functions in Step 3. 5. Verify whether the selected side in Step 1 is equal to the other side. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Trigonometry (continued) DRAW TRIGONOMETRIC SKETCH GRAPHS OF 𝒚𝒚 = 𝒂𝒂 𝒂𝒂𝒂𝒂𝒂𝒂 𝒃𝒃 + = 𝒅𝒅; 𝒃𝒃 𝒚𝒚 = 𝒂𝒂 𝒂𝒂𝒂𝒂𝒂𝒂 𝒃𝒃 + = 𝒅𝒅 AND 𝒚𝒚 = 𝒂𝒂 𝒂𝒂 𝒃𝒃 𝒂𝒂 𝒂𝒂 𝒃𝒃 + = 𝒅𝒅 for − 𝟐𝟐 ≤ 𝒃𝒃 𝟐𝟐 𝒙𝒙 ≤ 𝟐𝟐 𝟐𝟐 1. Sketch the basic trigonometric graph. 2. Identify the amplitude 𝑎𝑎 and modify the basic graph. 3. Identify the frequency 𝑏𝑏 and modify the graph in Step 2. 4. Identify the phase angle and modify the graph in Step 3. 𝑐𝑐 𝑑𝑑 5. Identify the vertical shift 𝑑𝑑 and modify the graph in Step 4. Downloaded 6. Modify the graph to fit from hlayiso.com the given domain. www.futuremanagers.com

Module 4: Trigonometry (continued) DRAW RECIPROCAL TRIGONOMETRIC SKETCH GRAPHS OF 𝒚𝒚 = 𝒄𝒄𝒄𝒄 𝒄𝒄𝒄𝒄 , 𝒚𝒚 = 𝒔𝒔 𝒄𝒄 𝒄𝒄 𝒔𝒔𝒔𝒔 𝒔𝒔 AND 𝒚𝒚 = 𝒄𝒄𝒄𝒄 𝒄𝒄𝒄𝒄 FOR − 𝟐𝟐 ≤ 𝟐𝟐 𝒙𝒙 ≤ 𝟐𝟐 𝟐𝟐 1. Write the reciprocal trigonometric equation in terms of sin𝑥𝑥, cos𝑥𝑥 or tan𝑥𝑥. 2. Compute points by means of the table method. 3. Create a Cartesian coordinate system. 4. Draw the reciprocal trigonometric graph and clearly show the asymptotes, and turning points and x-intercepts where applicable. Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Sketch graphs IDENTIFY CHARACTERISTICS IN RESPECT OF FUNCTIONS AND NON- FUNCTIONS • Domain and range; • Independent and dependent variable; • Functions and non-functions; • Points of symmetry with reference to an axis or the lines 𝑦𝑦 = ±𝑥𝑥; • Continuous and discontinuous functions or non-functions; and • Inverse functions and non-functions. Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Sketch graphs (continued) DRAW NEAT SKETCH GRAPHS OF FUNCTIONS AND NON-FUNCTIONS 𝑥𝑥 2 𝑦𝑦 2 • Straight line graph 𝑦𝑦 = 𝑚𝑚 + 𝑐𝑐; • Centred hyperbola 2 − 2 = 1; 𝑎𝑎 𝑏𝑏 𝑚𝑚 • Circle and semi-circle graph • Exponential and logarithmic graph 𝑦𝑦 = 𝑥𝑥 2 + 𝑦𝑦 2 = 𝑟𝑟 2 ; 𝑘𝑘𝑎𝑎𝑛𝑛 and 𝑦𝑦 = 𝑘𝑘𝑘𝑘𝑘𝑘𝑔𝑔𝑎𝑎 (𝑛𝑛 ); and 𝑛𝑛 𝑛𝑛 • Rectangular hyperbola x𝑦𝑦 = 𝑐𝑐; • Parabola 𝑦𝑦 = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏 + 𝑐𝑐. 𝑏𝑏 𝑥𝑥 2 𝑦𝑦 2 • Ellipse 2 + 2 = 1; 𝑎𝑎 𝑏𝑏 Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation APPLY THE THEOREMS ON LIMITS TO CALCULATE LIMITS USING ALGEBRAIC EXPRESSIONS AND QUOTIENTS Several different techniques are used to evaluate limits of determinate forms. The strategy for using limits is: 1. Evaluate limits by applying direct substitution. 0 ∞ 2. If Step 1 yields indeterminate forms or , then apply one of the 0 ∞ remaining techniques. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) GIVE THE BINOMIAL THEOREM IN GENERAL TERMS The general term of the binomial expression is: 𝑛𝑛 𝑛𝑛−1 𝑟𝑟 𝑇𝑇𝑟𝑟+1 = 𝑥𝑥 ℎ 𝑟𝑟 Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) APPLY THE BINOMIAL THEOREM WITH RATIONAL INDICES TO EXPAND A SIMPLE BINOMIAL TO FOUR TERMS Strategy for using the generalised version of the binomial theorem: 1. Write the expression in the form 𝑥𝑥 + ℎ 𝑛𝑛 . 2. Expand the expression using the generalised version of the binomial theorem. 3. Simplify the expansion. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) DEFINE DIFFERENTIATION AS RATE OF CHANGE AND DERIVE THE 𝒇𝒇 +∆ −𝒇𝒇 𝒇𝒇 +𝒉𝒉 −𝒇𝒇 EXPRESSION lim OR lim FROM FIRST 𝒙𝒙 𝒙𝒙 𝒙𝒙 𝒙𝒙 𝒙𝒙 ∆ → ∆ 𝒙𝒙 𝟎𝟎 𝒙𝒙 𝒉𝒉→ 𝟎𝟎 𝒉𝒉 PRINCIPLES WITH THE AID OF A SKETCH Strategy for determining the derivative from first principles: 1. Replace 𝑥𝑥 in the given function 𝑓𝑓(𝑥𝑥) with 𝑥𝑥 + ℎ, that is, 𝑓𝑓(𝑥𝑥 + ℎ). 2. Simplify the numerator, that is, 𝑓𝑓(𝑥𝑥 + ℎ)– 𝑓𝑓(𝑥𝑥). 𝑓𝑓 𝑥𝑥+ℎ −𝑓𝑓 𝑥𝑥 3. Remove the common factor ℎ and simplify by cancelling the ℎ. ℎ 4. Substitute ℎ =Downloaded 0 and simplify. from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) 𝒅𝒅 DETERMINE OF THE STANDARD FORMS: 𝒚𝒚 = 𝒌𝒌; 𝒚𝒚 = 𝒌𝒌 ; 𝒚𝒚 = 𝒌𝒌𝒂𝒂 ; 𝒅𝒅 𝒙𝒙 𝒏𝒏 𝒅𝒅 𝒙𝒙 𝒅𝒅 𝒚𝒚 = 𝒌𝒌 𝒆𝒆 𝒙𝒙 ; 𝒚𝒚 = 𝒌𝒌 ln 𝑘𝑘 ; 𝒚𝒚 = 𝒌𝒌 log 𝒂𝒂 ;𝒚𝒚 = 𝒌𝒌 𝑘𝑘 𝒙𝒙 𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌 ; 𝒚𝒚 = 𝒌𝒌 𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌 ; 𝒚𝒚 = 𝒌𝒌 𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌 ; 𝒚𝒚 = 𝒌𝒌 𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌 ; 𝒚𝒚 = 𝒌𝒌 𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌 , AND 𝒚𝒚 = 𝒌𝒌 𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌𝒌 . Strategy for determining derivatives of standard functions: 1. Rewrite the original function in the format of the standard differentiation formulas. 2. Differentiate the rewritten function. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) APPLY THE CHAIN RULE TO A FUNCTION OF A FUNCTION TO DETERMINE THE FIRST DERIVATIVES. 1. Substitute 𝑢𝑢 = 𝑛𝑛 for the composite functions, and rewrite the composite 𝑛𝑛 functions in terms of 𝑢𝑢. 𝑑𝑑𝑑𝑑 𝑑𝑑 2. Compute and . 𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑦𝑦 3. Multiply and to find . 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑 4. Rewrite the resulting derivative in terms of 𝑥𝑥 and simplify. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) APPLY THE DERIVATIVES TO FIND THE FIRST DERIVATIVES OF POLYNOMIALS Strategy for determining derivatives of polynomials: 1. Rewrite the derivative of a polynomial as separate derivatives by using the sum and/or difference rule for differentiation. 2. Differentiate Downloaded each individual function. from hlayiso.com www.futuremanagers.com

Module 6: Limits and differentiation (continued) APPLY THE PRODUCT AND QUOTIENT RULES FOR DIFFERENTIATION TO DIFFERENTIATE SIMPLE PRODUCTS AND QUOTIENTS Strategy for determining the Product rule and Quotient rule: 1. Select 𝑓𝑓(𝑥𝑥) and g(𝑥𝑥), and compute 𝑓𝑓′(𝑥𝑥) and g′(𝑥𝑥). 2. Substitute 𝑓𝑓(𝑥𝑥), g(𝑥𝑥), 𝑓𝑓′(𝑥𝑥) and g′(𝑥𝑥) into the Product or Quotient rule. 3. Simplify the Product rule or Quotient rule. Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Integration UNDERSTAND THE CONCEPT OF INTEGRATION AS A SUMMATION FUNCTION AND AS THE INVERSE OF DIFFERENTIATION Differentiation and integration (antiderivative) are inverse processes. The integration is denoted by an integral sign ∫ . Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Integration (continued) APPLY STANDARD FORMS OF INTEGRALS AS THE INVERSE OF DIFFERENTIATION Strategy for determining integrals of standard functions: 1. Rewrite the original integral in the standard integration formulas format. 2. Integrate the rewritten integral and include a constant of integration. 3. Verify the solution by differentiating. Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Integration (continued) INTEGRATE COMPOSITE FUNCTIONS Strategy for determining integrals of composite functions: 1. Substitute 𝑢𝑢 = 𝑛𝑛 for the composite functions 𝑘𝑘. 𝑎𝑎𝑛𝑛 , 𝑘𝑘. 𝑒𝑒 𝑛𝑛 and 𝑢𝑢 = 𝑏𝑏 for 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑏𝑏 the composite functions 𝑘𝑘. sin(𝑏𝑏 ) and 𝑘𝑘. cos(𝑏𝑏 ). 𝑏𝑏 𝑏𝑏 2. Compute 𝑑𝑑 = 𝑛𝑛. 𝑑𝑑 or 𝑑𝑑 = 𝑏𝑏. 𝑑𝑑 . and rewrite the integral in terms of 𝑢𝑢. 𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑑𝑑 3. Integrate in terms of 𝑢𝑢 and include the constant of integration. 4. Rewrite the resulting integral in terms of 𝑥𝑥 and verify the solution by differentiating. Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Integration (continued) INTEGRATE POLYNOMIALS Strategy for determining integrals of polynomials 1. Rewrite the integral of a polynomial as separate integrals by using the sum and/or difference rule for integration. 2. Integrate each individual integral. 3. Verify the solution by differentiating. Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Integration (continued) CALCULATE DEFINITE INTEGRALS OF FUNCTIONS The definite integral is an extension of the indefinite integral, that is, the evaluation of the antiderivative that yields a number. Strategy for determining definite integrals: 1. Rewrite the original integral in the standard integration formulas format. 2. Integrate the rewritten integral. 3. Evaluate the antiderivative, that is, the upper limit evaluation minus the lower limit evaluation. Downloaded from hlayiso.com www.futuremanagers.com

Module 7: Integration (continued) DETERMINE THE MAGNITUDE OF AN AREA INCLUDED BY A CURVE AND THE -AXIS, OR BY A CURVE, THE -AXIS AND THE ORDINATES 𝒙𝒙 𝒙𝒙 𝒙𝒙 = 𝒂𝒂 AND 𝒙𝒙 = 𝒃𝒃, WHERE 𝒂𝒂 AND 𝒃𝒃 ARE INTEGERS Strategy for determining area by means of integration: 1. Identify whether the shaded required area(s) are above the 𝑥𝑥-axis. 2. Indicate the representative strip and limits of integration. 3. Write down the area of representative strip, that is, ∆𝐴𝐴 = 𝑦𝑦. Δ𝑥𝑥. 4. Compute theDownloaded total area by usingfrom the definite integral. hlayiso.com www.futuremanagers.com